I lived in Richmond "before kids" and absolutely loved it - the amazing park, great vibe, the river, not too bad a commute (depending on where you are going). As it's such a beautiful place, it's quite expensive; also, where you live relative to the station (depending on your or your partner's commute) can be tricky. The airplane noise, however, is awful - if this is something that would really bother you (and that's both at home - and also every time you go outside and in Richmond Park), I would consider another area.
Within Richmond, another very popular area with families is Kew.
If the airplane noise bothers you, and perhaps you are on an expat package, then you could also consider other areas, such as:
a) Teddington / Hampton Wick - cheaper than Richmond, no airplane noise, a tiny bit further, but with Bushy Park on your doorstep
b) Wimbledon - again tricky around the station, Wimbledon Hill is beautiful and really expensive, but quite a hike from the station
c) Between the Commons, where many of us live from this site. It's a great area only in Zone 2, and usually you live near Clapham Junction (mainline station) or Clapham Common (Northern line tube), depending on where you are looking to commute to. "Between the Commons" means the area between Clapham and Wandsworth Common. There are lots of great schools, both private and state. If state, then the two best schools are Belleville and Honeywell, and both have catchment areas - but you will find plenty of properties for rent within catchment. If private, there can be quite the competition, although that's probably lessened in the post-Covid world, with many families deciding to go state instead. We do have airplane noise - but not as bad as Richmond - and we don't have an amazing park like Richmond Park - although Battersea Park is not that far.
Best of luck, and if you have any questions on Belleville / Honeywell catchments, system etc, feel free to PM me.